Theorem basis2 22101

Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
basis2	⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem basis2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasis2g 22098	. . . . 5 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧))))
2	1	ibi 266	. . . 4 ⊢ (𝐵 ∈ TopBases → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)))
3		ineq1 4139	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧))
4		sseq2 3947	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝑧)))
5	4	anbi2d 629	. . . . . . . . 9 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
6	5	rexbidv 3226	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
7	6	raleqbi1dv 3340	. . . . . . 7 ⊢ ((𝑦 ∩ 𝑧) = (𝐶 ∩ 𝑧) → (∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
8	3, 7	syl 17	. . . . . 6 ⊢ (𝑦 = 𝐶 → (∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧))))
9		ineq2 4140	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷))
10		sseq2 3947	. . . . . . . . . 10 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (𝑥 ⊆ (𝐶 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐶 ∩ 𝐷)))
11	10	anbi2d 629	. . . . . . . . 9 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
12	11	rexbidv 3226	. . . . . . . 8 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
13	12	raleqbi1dv 3340	. . . . . . 7 ⊢ ((𝐶 ∩ 𝑧) = (𝐶 ∩ 𝐷) → (∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
14	9, 13	syl 17	. . . . . 6 ⊢ (𝑧 = 𝐷 → (∀𝑤 ∈ (𝐶 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝑧)) ↔ ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
15	8, 14	rspc2v 3570	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) → ∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
16		eleq1 2826	. . . . . . . 8 ⊢ (𝑤 = 𝐴 → (𝑤 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
17	16	anbi1d 630	. . . . . . 7 ⊢ (𝑤 = 𝐴 → ((𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
18	17	rexbidv 3226	. . . . . 6 ⊢ (𝑤 = 𝐴 → (∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) ↔ ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
19	18	rspccv 3558	. . . . 5 ⊢ (∀𝑤 ∈ (𝐶 ∩ 𝐷)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))
20	15, 19	syl6com 37	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝑦 ∩ 𝑧)∃𝑥 ∈ 𝐵 (𝑤 ∈ 𝑥 ∧ 𝑥 ⊆ (𝑦 ∩ 𝑧)) → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))))
21	2, 20	syl 17	. . 3 ⊢ (𝐵 ∈ TopBases → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))))
22	21	expd 416	. 2 ⊢ (𝐵 ∈ TopBases → (𝐶 ∈ 𝐵 → (𝐷 ∈ 𝐵 → (𝐴 ∈ (𝐶 ∩ 𝐷) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))))))
23	22	imp43 428	1 ⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  TopBasesctb 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-bases 22096

This theorem is referenced by:  tgcl  22119  restbas  22309  txbas  22718  basqtop  22862  tgioo  23959